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JID:TCS AID:9537 /FLA Doctopic: Algorithms, automata, complexity and games [m3G; v 1.117; Prn:2/12/2013; 15:44] P.1 (1-18)Theoretical Computer Science ()

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i-criteria and approximation algorithms for restrictedatchings

onaldo Mastrolilli, Georgios Stamoulis

ituto Dalle Molle di Studi sullIntelligenza Articiale, IDSIA (USI/SUPSI), Manno-Lugano, Switzerland

r t i c l e i n f o a b s t r a c t

ywords:proximation algorithmsmbinatorial optimizationear programmingaph algorithms

In this work we study approximation algorithms for the Restricted matching problem whichis dened as follows: given a graph in which each edge e has a color ce and a protpe Q+, we want to compute a maximum (cardinality or prot) matching in which nomore than w j Z+ edges of color c j are present. This kind of problems, beside thetheoretical interest on its own right, emerges in multi-ber optical networking systems,where we interpret each unique wavelength that can travel through the ber as a colorclass and we would like to establish communication between pairs of systems. We studyapproximation and bi-criteria algorithms for this problem which are based on linearprogramming techniques and, in particular, on polyhedral characterizations of the naturallinear formulation of the problem. In our setting, we allow violations of the bounds w j andwe can model our problem as a bi-criteria problem: we have two objectives that we wantto optimize namely (a) to maximize the prot (maximum matching) while (b) minimizingthe violation of the color bounds. We prove how we can beat the integrality gap of thenatural linear programming formulation of the problem by allowing only a slight violationof the color bounds. In particular, our main result is constant approximation bounds forboth criteria of the corresponding bi-criteria optimization problem.

2013 Elsevier B.V. All rights reserved.

Introduction

Consider the following game: we organize a competition in a school and we have a set of binary games such as chess,O, tavli (a.k.a. backgammon), etc. Some pairs of students are interested in playing one particular game, whereas some otherirs are interested in some other game. We only have a limited amount of free boards for a particular game. Ideally, weould like to satisfy as many pairs of students as possible with the available amount of boards. This simple game capturesactly the essence of the problem of this article: we can formulate the above scenario as a graph G = (V , E), where Vthe set of students, and two students that are interested in a particular game (say chess) are connected with an edgea particular color (say black) associated with this game. Let w j N be the number of available boards of game j. Thene task of the organizers is to compute a maximum matching that uses at most w j boards of game j. Call this problemunded Color Matching problem.More formally, the problem can be stated as follows:

A preliminary version of this work appeared in Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO 2012), Lecture NotesComputer Science. Supported by the Swiss National Science Foundation Project No. 200020-122110/1 Approximation Algorithms for Machine Schedulingrough Theory and Experiments III and by Hasler Foundation Grant 11099.Corresponding author.E-mail addresses: monaldo@idsia.ch (M. Mastrolilli), georgios@idsia.ch (G. Stamoulis).

04-3975/$ see front matter 2013 Elsevier B.V. All rights reserved.tp://dx.doi.org/10.1016/j.tcs.2013.11.027

JID:TCS AID:9537 /FLA Doctopic: Algorithms, automata, complexity and games [m3G; v 1.117; Prn:2/12/2013; 15:44] P.2 (1-18)2 M. Mastrolilli, G. Stamoulis / Theoretical Computer Science ()

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anisgrthsownition 1 (Bounded Color Matching). We are given a (simple, un-directed) graph G = (V , E) with vertex set V and edget E such that |V | = n and |E| = m. The edge set is partitioned into k sets E1 E2 Ek i.e. every edge e has colorj if e E j and a prot pe Q+ . We are asked to nd a maximum (weighted) matching M (or a matching of maximumrdinality) such that in M there are no more that w j edges of color C j , where w j Z+ i.e. a matching M such that E j | w j , j [k].

In the following, we denote as C the collection of all the color classes. In other words, C = {C j} j[k] . Moreover, for aven edge e E(G), we denote by c(e) its color i.e. c(e) = C j e E j .Besides the previously mentioned toy problem, Bounded Color Matching emerges in optical networking systems: in antical ber we allow multiplexing of different frequencies (i.e. different beams of light can travel at the same time insidee same ber), but we have limited capacities of the number of light beams of a particular frequency that we allow to travelmultaneously through the system, due to potential interference problems. We would like to establish connections betweenmaximum number of (disjoint) pairs of systems while at the same time respecting the maximum number of connectionsing the same frequency we allow in multiplexing. Moreover, the Bounded Color Matching problem with 2 colors (i.e. these that each edge is colored either blue or red) can be used in approximately solving the Directed Maximum Routingd Wavelength Assignment problem (DirMRWA) [44] in rings which are fundamental network topologies, see [45] (also [8,for alternative and slightly better approximation algorithms and [3] for combinatorial algorithms). Here, approximatelylving means that an (asymptotic) -approximation algorithm for maximum bluered matching results in an (asymptotic)12 -approximation algorithm for DirMRWA in rings.

. History and related results

Characterizations and algorithms for maximum matchings in graphs have a very long history. One of the rst attemptscharacterize the structure of matchings was as early as the 1957 when Claude Berge characterized the structure of

aximum matchings with respect to alternating and augmenting paths [5]: a matching M on a given graph G is maximumand only if G contains no M-augmenting paths. A path that alternates between edges in M and edges not in M (forgiven matching M) is called an M-alternating path. An M-alternating path whose endpoints are unsaturated by M (i.e.rtices that do not have edges incident to them that are in M) is called an M-augmenting path. M-augmenting pathsovide a certicate of the non-maximality of M .Given this characterization of maximum matchings, an algorithm is immediate for computing a maximum matching Ma graph G:

Initialize M := while there exists an M-augmenting path Pdo augment M along P .

Of course the running time of the above general algorithm depends on how fast we can nd M-augmenting pathsa graph G with m edges and n vertices. In case the graph is bipartite nding maximum matchings can be done (rel-

ively) easily in time O(mn) [28,29], beating the trivial brute-force approach which simply enumerates all possible-augmenting paths which takes time O(nm). We remind that a graph G is bipartite if its vertex set can be partitionedto two sets V1, V2 such that every edge connects a vertex in V1 to one in V2; that is, V1 and V2 are independent sets.uivalently, a bipartite graph is a graph that does not contain any odd-length cycles.The case when G is not bipartite is signicantly more complicated because of the presence of odd-length cycles. In his

minal 1965 article, Jack Edmonds presented an O(n2m) time algorithm for solving the maximum matching problemgeneral graphs [17]. In fact, it was precisely this article that introduced the concept of polynomially time solvable prob-ms as tractable problems. As it always happens, the running time of this algorithm has been signicantly improveder the years. In [18], by a sophisticated use of some data structures, a running time of min{nm logn,n2.5} was shownr computing a maximum matching in a graph G , which was later improved to O(n2.5), see [40].The previous algorithms are purely combinatorial. Other very successful approaches for computing maximum matchingsgraphs are based on using of algebraic methods and/or randomization. We will not go into much detail here, exceptentioning the most important results, which include an O(n+1) time algorithm [47], and two O(n) time algorithms3] and [26] (see also [27]), where = inf{c: two n n matrices can be multiplied in time O(nc)} (we note that g2 7 2.38).All the above algorithms work for the un-weighted (uniform weights) case. In case we have a weight function w : E Q+d we want to compute a maximum weighted matching, then other techniques are required. The most common techniquethe so-called Hungarianmethod [32], which is a primaldual technique, initially introduced for bipartite graphs. For generalaphs, similar primaldual techniques have been employed, see for example [16,1921] among others. The idea, as most ofe primaldual schemata, is to build up feasible primal and dual solutions simultaneously and show that at the end bothlutions satisfy complementary slackness conditions and hence by the duality theorem, the primal solution is a maximumeight matching. Another approach for the maximum weight matching problem is to maintain a feasible matching and

JID:TCS AID:9537 /FLA Doctopic: Algorithms, automata, complexity and games [m3G; v 1.117; Prn:2/12/2013; 15:44] P.3 (1-18)M. Mastrolilli, G. Stamoulis / Theoretical Computer Science () 3

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ermcothlecoExmbofobyiny to successively augment it to increase its weight, until no more augmenting is possible, see for example [15]. Formprehensive accounts of the matching problem, we refer to [37] and [49].

2. Constrained matching problems

Since the task of computing maximum matchings is an extremely well studied and basic problem, the interest hasifted towards some other versions of maximum matchings, in particular to versions where we seek a maximum matchingbject to additional criteria (constraints). These criteria reduce the feasible solution space, making it (usually) harder tompute optimal solutions in polynomial time. In this direction, Bounded Color Matching problems were studied as earlythe 1970s as a very interesting generalization of the classical maximum matching problem: In [22], the problem wasned as Multiple Choice Matching (reference problem [GT55]) and proved to be NP-complete even for the very special casehere the graph is bipartite, each color class contains at most 2 edges (i.e. |E j| 2, j) and w j = 1, C j C . This problem,ds numerous practical applications, from classroom scheduling to image segmentation among others, see also [30,31].Moreover, the uniform weight version of Bounded Color Matching problem is also closely related to the Labeled Matchingoblem [42,11] in which all bounds w j are set equal to 1, i.e. we would like a maximum matching with at most one edger color. In [42] it was proven that even the very special case of 2-regular bipartite graphs where each color appears twice.e. in at most two edges), the problem is APX-hard and so a PTAS is immediately out of reach for Bounded Color Matchingee also [41]).Budgeted versions of the maximum matching problem, have been recently studied intensively. Here, by budgeted versiona combinatorial optimization problem we mean the following: Besides the prot function p : F Q+ associated

ith every feasible solution F F for (where F is the set of all feasible solutions for ), we are also given a set of st functions {i}i[] such that i :F Q+ , and for every cost function i a budget i Q+ . The budgeted optimizationrsion of , which we call b , can be then formulated as follows (assuming that is a maximization problem):

max p(F ), subject to F F and i(F ) i, i []. (1)In [24] (see also [25]) the authors considered the 2-budgeted maximum matching problem (i.e. the case where = 2)d devised a PTAS. This algorithm works roughly as follows: First of all, a guessing step is performed that guesses the most valuable edges of the optimal matching. Then, an optimal fractional matching x is computed for the rest of theaph (for example by solving the linear programming relaxation of the problem). By Carathodorys theorem [10], x canwritten as a convex combination of at most three (possibly unfeasible) matchings i.e. x = 1x1 + 2x2 + (1 1 2)x3.en, the algorithm consists of two merging steps: in the rst step, given the rst two matchings x1 and x2 the output is aird matching z with comparable prot and which is not costlier than x1 + (1)x2 for = 11+2 with respect to bothe two extra cost functions. Then the same procedure is again applied to z and x3 with parameter = 1+21+2+(112) , i.e.e merge x3 with z such that the new matching z is feasible (with respect to both cost functions) and (almost) optimal.This was further improved in [13] where it is given a PTAS for a xed number of budgets. The authors there providedth randomized and deterministic PTASs for the problem. The randomized version is based on strong concentration boundssome suitable martingale processes (see also [12] for some closely related results and techniques). The deterministic PTASn be seen as a bi-criteria approximation, and the nal solution returned is within (1 ) the optimal but it might violatee budgets by a factor of (1 + ) (i.e. the solution z returned has the property that ci(z) (1 + )i , i). Moreover, forbounded number of budgets the authors prove an almost optimal approximation guarantee, but with allowing a veryrge (i.e. logarithmic) overow on the budgets (as before, this means that for the computed solution z, z has the propertyat ci(z) i logi , budget i). These results generalize the results for the budgeted bipartite matching problem, for whichPTAS was known for the case of one budget [6,7], or in the case of xed number of budgets [23] in which a (1 ,1+ )-criteria approximation was shown.To the best of our knowledge, the rst case where matching problems with cardinality (disjoint) budgets were consid-

ed, was in [45] where the authors dened and studied the bluered Matching problem: compute a maximum (cardinality)atching that has at most w blue and at most w red edges, in a bluered colored (multi)-graph. A 34 polynomial timembinatorial approximation algorithm and an RNC2 algorithm were presented (that computes the maximum matchingat respects both budget bounds with high probability). We note that the exact complexity of the bluered matching prob-m is not known: it is only known that bluered matching is at least as hard as the Exact Matching problem [46] whosemplexity is open for more than 30 years. A polynomial time algorithm for the bluered matching problem will imply thatact Matching is polynomial time solvable. On the other hand, bluered matching is probably not NP-hard since it ad-its an RNC2 algorithm. We note that this algorithm ca...